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Consider a system of equations of implicit function type, F(x,y)=0. Here F(x,y) is a
vector of analytic/algebraic power series.
(Artin) Any formal solution y(x) of this system is approximated (x-adically) by
solutions in analytic/algebraic series.
Geometrically, suppose a morphism of (analytic/Nash) scheme-germs admits a formal
section. This formal section is adically approximated by analytic/Nash sections.
(The inverse question of Grothendieck) Given a map of (analytic/Nash) scheme-germs. Suppose its formal stalk is a section of some formal morphism. Is the initial map a section of some (analytic/Nash) morphism? The answer is yes in the Nash case (Popescu) and no in the analytic case (Gabrielov). The left-right version of this question is important for the study of morphisms of scheme-germs, and was addressed by M.Shiota in the real-analytic/Nash context. These versions appear to be particular cases of the general "Artin approximation problem on quivers". I will present the characteristic-free results. |